Solve (x)=2x^2-5/2x^2+1

$$( x ) = \frac { 2 x ^ { 2 } - 5 } { 2 x ^ { 2 } + 1 }$$

Solve for x

$$x=-1$$

Solution Steps

Subtract \frac{2x^{2}-5}{2x^{2}+1} from both sides.

$$x-\frac{2x^{2}-5}{2x^{2}+1}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{2x^{2}+1}{2x^{2}+1}.

$$\frac{x\left(2x^{2}+1\right)}{2x^{2}+1}-\frac{2x^{2}-5}{2x^{2}+1}=0$$

Since \frac{x\left(2x^{2}+1\right)}{2x^{2}+1} and \frac{2x^{2}-5}{2x^{2}+1} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\left(2x^{2}+1\right)-\left(2x^{2}-5\right)}{2x^{2}+1}=0$$

Do the multiplications in x\left(2x^{2}+1\right)-\left(2x^{2}-5\right).

$$\frac{2x^{3}+x-2x^{2}+5}{2x^{2}+1}=0$$

Multiply both sides of the equation by 2x^{2}+1.

$$2x^{3}+x-2x^{2}+5=0$$

Rearrange the equation to put it in standard form. Place the terms in order from highest to lowest power.

$$2x^{3}-2x^{2}+x+5=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 5 and q divides the leading coefficient 2. List all candidates \frac{p}{q}.

$$±\frac{5}{2},±5,±\frac{1}{2},±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$x=-1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 2x^{3}-2x^{2}+x+5 by x+1 to get 2x^{2}-4x+5. Solve the equation where the result equals to 0.

$$2x^{2}-4x+5=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 2 for a, -4 for b, and 5 for c in the quadratic formula.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 2\times 5}}{2\times 2}$$

Do the calculations.

$$x=\frac{4±\sqrt{-24}}{4}$$

Since the square root of a negative number is not defined in the real field, there are no solutions.

$$x\in \emptyset$$

List all found solutions.

$$x=-1$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{6}i}{2}+1\approx 1-1.224744871i$$

$$x=-1$$

$$x=\frac{\sqrt{6}i}{2}+1\approx 1+1.224744871i$$